Show Region of Integration for Multiple Integrals

[Brewer]

How do I go about showing the region of integration represented by a repeated integral? (Just for 2 dimensional functions)

All the diagrams I've seen show the area under the graph, but with like 2 bars across it. I don't understand what all this represents and my poor quality notes are of absolutely no help whatsoever.

 

[matt grime]

'Showing' in what sense? Drawing on paper? Writing it as formulae in the integral sign?

 

[Brewer]

Sorry, yes drawing.

Sketching would probably have been a better way of phrasing it.

 

[HallsofIvy]

A double integral must look something like this:
[tex]\int_{x=a}^b\int_{y=f(x)}^{g(x)} u(x,y)dy dx[/itex]
That is, the limits on the outer integral must be numbers, not functions or x or y, and the limits on the inner integral must be functions of the "outer" variable (x in this case because the "outer" integral is with respect to x), not the "inner" variable (y in this case).

The limits of integration tell you "x can take on all values between a and b and for each x y goes from f(x) up to g(x)". Draw two vertical lines at x= a and x= b to show that boundary. Now draw the graphs of y= f(x) and y= g(x). The region bounded by those 4 lines/curves is the area of integration. (In many problems, it happens that the two curves intersect precisely at x= a and/or x= b.)

 

[matt grime]

Well, you draw it and indicate however you want which part of the diagram is the bit you're integrating over. Often cross hatching is the preferred way but there really is no golden rule.